CN102710288B - Orthogonal PSWF (Prolate Spheroidal Wave Function) pulse design method on basis of cross-correlation matrix diagonalization - Google Patents

Abstract

The invention discloses an orthogonal PSWF (Prolate Spheroidal Wave Function) pulse design method on the basis of the cross-correlation matrix diagonalization, which is the orthogonal PSWF pulse design method. The orthogonal PSWF pulse design method comprises the following steps of: by calculating a cross-correlation matrix of a PSWF pulse which participates in the orthogonalization design, carrying out diagonalization transformation on the cross-correlation matrix to obtain an orthogonal matrix; and then carrying out matrix multiplication operation by a transposed matrix of the orthogonal matrix and the PSWF pulse so as to implement the orthogonalization of the PSWF pulse. According to the method, the calculating time complexity and the storage space of the orthogonal PSWF pulse design are reduced; and the method is not only suitable for the orthogonal design of the base band PSWF pulse, but also suitable for the orthogonal design of the band pass PSWF pulse.

Description

A kind of based on the diagonalizable orthogonal PSWF pulse design method of cross-correlation matrix

Technical field

The present invention relates to the waveform design method in radio communication, particularly relate to a kind of time domain orthogonal pulse design method.

Background technology

Elliptically spherical function collection (Prolate Spheroidal Wave Functions, PSWF) there is time and frequency zone energy accumulating the best, time domain biorthogonal, complete, the approximate good characteristics such as time limit band is limit, frequency spectrum is controlled, at the beginning of proposing, just receive the extensive concern of academia, and show good application prospect.First Slepian and Pollak etc. propose and have studied PSWF function set in the Bell laboratory research report of 1961, and in the time of 20 years thereafter, a series of correlative study report of consecutive publications.

About the form of Definition of PSWF, mainly contain following two kinds.

(1) differential equation definition

The differential equation definition of PSWF is as follows:

$[{\left(\frac{T}{2}\right)}^2-t^2]\frac{d^2{\mathrm{\ψ}}_n(c,t)}{d{t}^2}-2t\frac{d{\mathrm{\ψ}}_n(c,t)}{\mathrm{dt}}+[{\mathrm{\χ}}_n\left(c\right)-c^2{\left(\frac{2t}{T}\right)}^2]{\mathrm{\ψ}}_n(c,t)=0,-T/2\≤t\≤T/2---\left(1\right)$

Wherein, ψ _n(c, t) is n rank PSWF, χ _nfor n rank PSWF characteristic of correspondence value, c is the time-bandwidth product of PSWF, and T is the duration width of PSWF.

(2) integral equation definition

${\∫}_{-T/2}^{T/2}{\mathrm{\ψ}}_n(c,\mathrm{\τ})h(t-\mathrm{\τ})\mathrm{d\τ}={\mathrm{\λ}}_n{\mathrm{\ψ}}_n(c,t)---\left(2\right)$

Wherein, the kernel function that h (t) is PSWF, λ _nn rank PSWF ψ _nthe encircled energy factor of (c, t).

If 1. kernel function h (t) has perfect low pass characteristic, i.e. h (t)=sin Ω t/ π t, Ω is angular frequency, then its frequency domain characteristic is:

$H\left(\mathrm{\&omega;}\right)=\left\{\begin{array}{cc}1,& \left|\mathrm{\&omega;}\right|<\mathrm{\&Omega;}\\ 0,& \mathrm{else}\end{array}\right.---\left(3\right)$

Then the integral equation of PSWF is defined as:

${\&Integral;}_{-T/2}^{T/2}{\mathrm{\&psi;}}_n(c,\mathrm{\&tau;})\frac{\sin \mathrm{\&Omega;}(t-\mathrm{\&tau;})}{\mathrm{\&pi;}(t-\mathrm{\&tau;})}\mathrm{d\&tau;}={\mathrm{\&lambda;}}_n{\mathrm{\&psi;}}_n(c,t)---\left(4\right)$

Now, be [-Ω ,+Ω] between the frequency domain energy accumulation regions of PSWF, therefore, usually the PSWF that formula (4) defines be called base band PSWF.

2. correspond, if h (t) has desirable bandpass characteristics, namely kernel function is:

H (t)=sin Ω _ht/ π t-sin Ω _lt/ π t (5) its frequency domain characteristic is:

$H\left(\mathrm{\&omega;}\right)=\left\{\begin{array}{cc}1,& {\mathrm{\&Omega;}}_L<\left|\mathrm{\&omega;}\right|<{\mathrm{\&Omega;}}_H\\ 0,& \mathrm{else}\end{array}\right.---\left(6\right)$

Then the integral equation of PSWF is defined as:

${\&Integral;}_{-T/2}^{T/2}{\mathrm{\&psi;}}_n(c,\mathrm{\&tau;})[\frac{\sin {\mathrm{\&Omega;}}_H(t-\mathrm{\&tau;})}{\mathrm{\&pi;}(t-\mathrm{\&tau;})}-\frac{\sin {\mathrm{\&Omega;}}_L(t-\mathrm{\&tau;})}{\mathrm{\&pi;}(t-\mathrm{\&tau;})}]\mathrm{d\&tau;}={\mathrm{\&lambda;}}_n{\mathrm{\&psi;}}_n(c,t)---\left(7\right)$

Now, be [Ω between the frequency domain energy accumulation regions of PSWF _l, Ω _h], therefore, usually the PSWF that formula (7) defines is called the logical PSWF of band.

At present, PSWF is used widely in multiple field.This function is used to small echo signal analysis, can reach the temporal resolution higher than sinc function; For digital picture and Digital Signal Processing, effectively can solve temporal resolution and spatial resolution contradiction, for Optical system modeling, there is better versatility; For radio communication channel research, can modeling high-speed mobile communications channel and time-frequency Selective Fading Channel more accurately; For signal of communication design, spectrum control flexibly can be realized, and there is higher spectrum efficiency and power efficiency.

But PSWF does not exist analytic solutions, method of value solving is usually adopted to obtain its approximate solution.For in the approximate solution method of PSWF, mainly contain Parr numerical solution (see document: Parr B, Cho B, Wallace K.A novel ultra-wideband pulse design algorithm [J] .IEEE Communication Letters, 2003,7 (5): 219-221.) method for solving, is reconstructed (see document: Khare K, George N.Sampling theory approach to prolate spheroidal wavefunctions [J] .Journal of Physics, 2003,36 (39): 10011-10021.) etc.These method for solving all comprise the process of solution matrix characteristic vector, thus there is the problem that implementation complexity is high.King's red magnitude at document " based on Legendre polynomial approximation method " (see document: Wang Hongxing, Chen Zhaonan, Zhao Zhi is brave. based on the ellipsoidal surface wave impulse method for designing [J] of Legnedre polynomial. and electric wave science journal, 2012,27 (1): 191-197.), in, a kind of low complex degree approximate solution method is proposed for base band PSWF.The method, according to the coefficient of PSWF calculation of parameter normalization Legnedre polynomial, obtains the numerical approximation solution of PSWF by the weighted sum of normalization Legnedre polynomial.

In order to effectively improve power efficiency and the spectrum efficiency of communication system, patent of invention (application number 200810237849.5) discloses the overlapping orthogonal pulses method for designing of a kind of time domain orthogonal, radio frequency channel, designed orthogonal pulses both can make communication system have higher band efficiency, had again good energy accumulating simultaneously.In the patent, be multiple mutually overlapping sub-bands by whole Pulse Design frequency band division, each sub-band utilize respectively Parr numerical solution solve the PSWF pulse of each rank, obtain orthogonal PSWF pulse finally by Schmidt orthogonalization method.Because the method has complexity and memory space higher computing time, thus improve the requirement to level of hardware.

Summary of the invention

In order to overcome the defect of prior art, the present invention is open a kind of based on the diagonalizable orthogonal PSWF pulse design method of cross-correlation matrix.The method by carrying out diagonalization conversion to the cross-correlation matrix of the PSWF pulse participating in orthogonalization design, thus realizes orthogonal PSWF Pulse Design.

In the present invention, orthogonalization for base band PSWF designs, on the basis of " the ellipsoidal surface wave impulse method for designing based on Legnedre polynomial ", normalization Legendre (Legendre) approximation by polynomi-als is adopted to solve the numerical approximation solution of PSWF, and the numerical solution of PSWF and orthogonalization procedure are optimized integrate process, to reduce implementation complexity.For the orthogonalization design of the logical PSWF of band, the present invention adopts and carrys out alternative Schmidt orthogonalization method based on cross-correlation matrix diagonalization.The method effectively can reduce complexity and memory space complexity computing time.

Design two aspects from the design of base band PSWF orthogonal pulses and the logical PSWF orthogonal pulses of band respectively below, elaborate technical measures of the present invention, to realize order of the present invention.

(1) base band PSWF orthogonal pulses design

The present invention is by analyzing based on the PSWF pulse derivation algorithm of normalization Legendre approximation by polynomi-als with based on the diagonalizable orthogonalization method of cross-correlation matrix, the numerical approximation of PSWF is solved and is optimized merging with orthogonalization procedure, establish the direct corresponding relation between orthogonal PSWF pulse and Legendre multinomial, thus the weighted sum matrix achieved by calculating orthogonal PSWF pulse, can directly obtain, by the orthogonal PSWF pulse of normalization Legendre approximation by polynomi-als, reducing implementation complexity.

1. based on the PSWF pulse derivation algorithm of normalization Legendre approximation by polynomi-als

From document " the ellipsoidal surface wave impulse method for designing based on Legnedre polynomial ", based on normalization Legendre polynomial expansion, jth rank PSWF can be expressed as:

${\mathrm{\&psi;}}_j(c,t)=\underset{k=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_k^j\&CenterDot;{\stackrel{\&OverBar;}{P}}_k\left(t\right),j=\mathrm{0,1},...,M---\left(8\right)$

Wherein, for kth rank normalization Legendre multinomial, its coefficient vector β ^jfor the characteristic vector of matrix A, namely

A β ^j=χ _jβ ^j(9) wherein, matrix A is defined as follows:

$A_{k,k+2}=\frac{(k+2)(k+1)}{(2k+3)\sqrt{(2k+5)(2k+1)}}\&CenterDot;c^2---\left(10\right)$

$A_{k,k}=k(k+1)+\frac{2k(k+1)-1}{(2k+3)(2k-1)}\&CenterDot;c^2---\left(11\right)$

$A_{k+2,k}=\frac{k(k-1)}{(2k-3)\sqrt{(2k-3)(2k+1)}}\&CenterDot;c^2---\left(12\right)$

Other element is 0.With ψ=[ψ ₀(c, t), ψ ₁(c, t) ..., ψ _m-1(c, t)] ^trepresent the vector become by M rank PSWF, B=[β ¹, β ²..., β ^m] represent Legendre multinomial weighting coefficient matrix, represent normalization Legendre polynomial vector, then formula (8) can be expressed as matrix multiple form:

ψ＝BP (13)

2. based on the diagonalizable orthogonalization method of cross-correlation matrix

For multiple PSWF pulse, the cross-correlation matrix of its pulse, as a kind of statistical indicator, can the most intensively reflect this correlation.Start with from its cross-correlation matrix exactly based on cross-correlation matrix diagonalization, eliminate the correlation between different pulse, realize the orthogonalization of pulse.One group of coherent signal can eliminate the correlation between each component to a certain extent after orthogonal transform.Cross-correlation matrix diagonalization is a kind of orthogonal transform be based upon on cross-correlation matrix basis, and meanwhile, from the performance eliminating each component correlations completely, cross-correlation matrix diagonalization is best.

Suppose that PSWF pulse is by the overlapping PSWF pulse ψ of M frequency spectrum _i(c, t) forms, then the cross-correlation matrix C of this pulse is:

Wherein, c _i,jbe the cross-correlation function of i-th pulse and a jth pulse, namely cross-correlation matrix C is symmetrical matrix, then must have orthogonal matrix X, make X ^tcX=Λ, the diagonal matrix that wherein Λ is is diagonal element with the M of a C characteristic value, X is the transformation matrix of cross-correlation matrix diagonalization.Make X=[x ₁, x ₂..., x _m], x _ifor the characteristic vector of cross-correlation matrix C, then have

$x_i^{T}C{x}_i={\mathrm{\&eta;}}_i---\left(15\right)$

Wherein, η _ifor i-th characteristic value of C.Make transformation factor x _i=[x _i1, x _i2..., x _iM] ^t, cross-correlation matrix diagonalization conversion is carried out to M PSWF pulse, the new pulse ψ ' obtained _i(c, t) is

${\mathrm{\&psi;}}_i^{\&prime;}(c,t)=\underset{k=1}{\overset{M}{\mathrm{\&Sigma;}}}{x}_{\mathrm{ik}}{\mathrm{\&psi;}}_k(c,t)---\left(16\right)$

New pulse ψ ' _i(c, t) and ψ ' _jthe cross-correlation function of (c, t) is

${\&Integral;}_{-T/2}^{T/2}{\mathrm{\&psi;}}_i^{\&prime;}(c,t){\mathrm{\&psi;}}_j^{\&prime;}(c,t)\mathrm{dt}=\underset{n=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{m=1}{\overset{M}{\mathrm{\&Sigma;}}}{x}_{\mathrm{in}}{x}_{\mathrm{jm}}{\&Integral;}_{-T/2}^{T/2}{\mathrm{\&psi;}}_m(c,t){\mathrm{\&psi;}}_n(c,t)\mathrm{dt}=\underset{n=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{m=1}{\overset{M}{\mathrm{\&Sigma;}}}{x}_{\mathrm{in}}{x}_{\mathrm{jm}}{c}_{\mathrm{mn}}---\left(17\right)$

According to X ^tcX=Λ, can obtain

${\&Integral;}_{-T/2}^{T/2}{\mathrm{\&psi;}}_i^{\&prime;}(c,t){\mathrm{\&psi;}}_j^{\&prime;}(c,t)\mathrm{dt}=\left\{\begin{array}{cc}{\mathrm{\&eta;}}_i& i=j\\ 0& i\&NotEqual;j\end{array}\right.---\left(18\right)$

Formula (18) demonstrates and converts through cross-correlation matrix diagonalization the new pulse obtained is mutually orthogonal.On this basis, the orthogonal new pulse obtained is normalized, namely obtains orthonormal PSWF pulse.

3. the numerical approximation of PSWF solves and orthogonalization procedure integration processing method

With ψ '=[ψ ' ₀(c, t), ψ ' ₁(c, t) ..., ψ ' _m-1(c, t)] ^trepresent orthogonal PSWF vector, then the PSWF pulsed orthogonal process represented by formula (16), can be expressed as the form of matrix multiple:

ψ′＝X ^Tψ (19)

The PSWF solution procedure of formula (13) is substituted into formula (19), can obtain:

ψ′＝X ^TBP (20)

Make D=X ^tb, the normalization Legendre fitting of a polynomial form that can obtain orthogonal PSWF pulse is:

ψ′＝DP (21)

Here title D is the weighted sum matrix of orthogonal PSWF pulse, uses d _irepresent the i-th row of this matrix D, then i-th orthogonal PSWF pulse can be expressed as

${\mathrm{\&psi;}}_i^{\&prime;}(c,t)=\underset{k=1}{\overset{N}{\mathrm{\&Sigma;}}}{d}_{\mathrm{ik}}{\stackrel{\&OverBar;}{P}}_k\left(t\right)---\left(22\right)$

Known by analysis mode (22), method disclosed by the invention, directly can set up orthogonal PSWF pulse and the polynomial corresponding relation of Legendre, simplify design procedure.Therefore, orthogonal pulses method for designing disclosed in this invention, being optimized integration process by the numerical approximation of PSWF being solved with orthogonalization procedure, reducing implementation complexity, thus improve the design efficiency of orthogonal PSWF pulse.

(2) the orthogonal pulses method for designing of the logical PSWF of band

Patent of invention (application number 200810237849.5) discloses the overlapping orthogonal pulses method for designing of a kind of time domain orthogonal, radio frequency channel.In the patent, the spectrum width that channel of communication and pulse take is divided into multiple bandwidth identical and mutually overlapping 50% wavelet road, each wavelet road utilizes respectively Parr numerical solution solve the numerical approximation solution of each rank PSWF, obtain orthogonal PSWF pulse finally by Schmidt orthogonalization method.But from the characteristic of Schmidt orthogonalization method, along with the increase participating in orthogonal design pulse number, this orthogonalization method can reduce the orthogonal performance of designed pulse, during for transmitting work and rest, makes the interference free performance of system decline.

Disclosed in this invention based on the diagonalizable orthogonalization method of cross-correlation matrix, the Schmidt orthogonalization procedure in patent of invention (application number 200810237849.5) is replaced, to improve designed orthogonal pulses performance.For the orthogonal pulses design of the logical PSWF of band, method for designing disclosed in this invention comprises the following steps: 1. radio frequency channel divide, 2. optimum configurations, 3. build characterisitic function and integral equation, 4. solving equation, 5. based on the diagonalizable orthogonalization of cross-correlation matrix.

1. radio frequency channel divides

Spectrum width channel of communication and orthogonal pulses taken is divided into the identical and mutually overlapping wavelet road of k bandwidth with even overlap mode;

2. optimum configurations

The parameter of orthogonal pulses is set, the spectrum width B taken by orthogonal pulses, frequency spectrum lower frequency limit f _lwith frequency spectrum upper limiting frequency f _h, to divide number k, the time-bandwidth product factor c of elliptically spherical function, the overlapping degree of frequency spectrum be that ρ determines in wavelet road;

3. characterisitic function and integral equation is built

Build characterisitic function and integral equation for each wavelet road, constructed characterisitic function is h _k(t)=2f _k,Hsinc (2f _k,Ht)-2f _k,Lsinc (2f _k,Lt), wherein, f _k,L, f _k,Hbe respectively lower-frequency limit and the upper limit of a kth sub-radio frequency channel, constructed integral equation is wherein, λ _kcharacterisitic function h when the numerical algorithm decomposed for feature based value solves integral equation _kthe characteristic value of (t) constructed matrix, ψ _k(c, t) is λ _kcorresponding characteristic function, the t time variable that to be interval be [-T/2, T/2];

4. solving equation is the integral equation that the Parr numerical algorithm decomposed by feature based value solves constructed by each wavelet road respectively, and by getting the characteristic function corresponding to a front m eigenvalue of maximum, obtains the elliptically spherical function pulse in each wavelet road;

5. referring to the cross-correlation matrix by calculating each wavelet road PSWF pulse participating in orthogonalization design based on the diagonalizable orthogonalization of cross-correlation matrix, this cross-correlation matrix being carried out diagonalization conversion, thus achieves the orthogonalization of PSWF pulse.

5. elaborate in the design of this patent base band PSWF pulsed orthogonal based on the diagonalizable orthogonalization of cross-correlation matrix about step, repeat no more here.

Compared with prior art, the present invention has following beneficial effect:

1. computing time, complexity was low

Suppose to need to carry out orthogonalization to M PSWF pulse, the sampling number of each PSWF pulse is K.Utilizing Legendre approximation by polynomi-als to solve in PSWF functional procedure, the polynomial sampling number of each rank Legendre is also K, and to each PSWF pulse, its Legendre multinomial number needed is wherein expression rounds up, and c is the time-bandwidth product of PSWF, and e is the end of natural logrithm.Construct an orthogonal PSWF pulse to need to be weighted summation to N number of Legendre multinomial, therefore, need to carry out NK computing (mainly multiplying), thus total operand of whole process is MNK.

For in the method for designing disclosed in patent (application number 200810237849.5), time complexity mainly concentrates on PSWF pulse and solves and Schmidt orthogonalization two steps.Based in the pulse solution procedure of Parr algorithm, operand comes from solution matrix characteristic vector process.When adopting classical Jacobi method to solve characteristic vector, tie up matrix to a K × K, operand is about K ³.First Schmidt method calculates weight coefficient a _i,j, be secondly weighted sum process.Generate i-th orthogonal pulses to need to calculate i-1 weight coefficient, M orthogonal pulses needs to calculate (M-1) M/2 weight coefficient altogether.Calculate a weight coefficient and need 2K+1 computing, total operand of weight coefficient computational process is (M-1) (2K+1) M/2.For weighted sum process, need (i-1) K multiplying to i-th new pulse, all pulse summation process operands are (M-1) MK/2.The operand of whole Schmidt orthogonal process is (M-1) (3K+1) M/2.The time complexity of patent (application number 200810237849.5) is: C _t=(M-1) (3K+1) M/2+K ³.

When producing pulse with Parr algorithm, suppose to sample with the sampling rate of 4 times of signal bandwidths, then the sampling number of pulse needs according to the method for designing disclosed in patent (application number 200810237849.5), the pulse number carrying out orthogonalization design in each subband is based on above condition, the time complexity of two kinds of orthogonal PSWF pulse design method compares analysis, as shown in Figure 1.As can be seen from the simulation result of Fig. 1, time complexity of the present invention will be significantly less than the method in patent (application number 200810237849.5), and along with the increase of time-bandwidth product, its time complexity is substantially constant, and the time complexity of patent (application number 200810237849.5) increases fast.

2. memory space is calculated little

Space complexity from FPGA hardware implementing process, main storage of variables of the present invention is Legendre multinomial coefficient and fit metric D.Multinomial group need altogether store N × K data, matrix D be M × N tie up, space complexity S in this way ₁for:

S ₁＝(M+K)N (23)

For patent (application number 200810237849.5), the eigenmatrix constructed by PSWF computational process is K × K dimension, and M the PSWF pulse obtained needs with M × K data representation.In Schmidt orthogonal process, need storage two PSWF pulses.When carrying out pulsed orthogonal, the space shared by eigenmatrix in last pulse solution procedure can be discharged, simultaneously because the sampling number of each pulse is far longer than pulse number, i.e. K>>M, thus compared with pulse solution procedure, the space complexity of Schmidt process can be ignored, and the space complexity of patent (application number 200810237849.5) determines primarily of pulse solution procedure, and its expression formula is:

S ₂＝(K+M)K (24)

When c span is [0,40], requisite space complexity is as shown in Figure 2 in FPGA hardware implementing process for two kinds of pulse design method.

As can be seen from the contrast of space complexity in Fig. 2, the data volume stored is needed obviously to reduce when adopting the present invention.Along with the increase of time-bandwidth product, space complexity recruitment of the present invention also will be significantly less than patent (application number 200810237849.5).When time-bandwidth product is 40, space complexity of the present invention is less than the half of patent (application number 200810237849.5).From space complexity, the present invention has larger advantage.

3. applied widely

In the method for designing disclosed in patent (application number 200810237849.5), Schmidt orthogonalization method is adopted to realize orthogonal PSWF Pulse Design, from the characteristic of Schmidt orthogonalization procedure, the requirement of this orthogonalization procedure is harsher, and it requires that the PSWF pulse participating in orthogonalization design is that Strict linear has nothing to do.。

And diagonalization of correlation matrices method disclosed in this invention, just linear combination is carried out to former PSWF pulse, the reconstruct namely in mathematical meaning, radical change has not been carried out to PSWF pulse.Meanwhile, the method is not strict with for the cross correlation of pulse, as long as be not linear relationship between each pulse of former pulse, namely the method is suitable for.Therefore, the method disclosed in the present applicability is wider.

Accompanying drawing explanation

Fig. 1 is the time complexity comparison diagram of the present invention and prior art two kinds of methods.

Fig. 2 is the present invention and the space complexity comparison diagram needed for prior art two kinds of methods.

Fig. 3 is that base band quadrature PSWF impulse waveform disclosed in this invention produces flow chart.

Fig. 4 is that band disclosed in this invention leads to orthogonal PSWF impulse waveform generation flow chart.

Embodiment

Below in conjunction with accompanying drawing, the present invention is described in further detail.

(1) base band PSWF orthogonalization design

For the design of base band PSWF orthogonalization, based on the diagonalizable orthogonal PSWF pulses generation flow process of cross-correlation matrix as shown in Figure 3, pulses generation can be carried out as follows.

1. sub-band division and optimum configurations

First, the spectrum width that channel of communication and pulse take is divided into k bandwidth identical and mutually overlapping 50% sub-band, k be greater than 0 positive integer; Then the parameter of orthogonal PSWF pulse is set, the spectrum width B taken by orthogonal pulses, frequency spectrum lower frequency limit f _lwith frequency spectrum upper limiting frequency f _h, wavelet road divide number k, elliptically spherical function time-bandwidth product factor c determine, spectrum width B and frequency spectrum lower frequency limit f _lwith frequency spectrum upper limiting frequency f _hthree meets relational expression: B=f _h-f _l, each sub-wave channel bandwidth is B ₀=2B/ (k+1), time-bandwidth product factor c and pulse duration T _s, each wavelet road bandwidth B ₀three meets relational expression: c=π B ₀t _s, the elliptically spherical function pulse number in each wavelet road is

2. fit metric D is determined

According to PSWF burst length bandwidth product factor c, by formula determine Legendre multinomial number needed for PSWF pulse, wherein expression rounds up, and c is the time-bandwidth product of PSWF, and e is the end of natural logrithm;

Setting up eigenmatrix according to formula (9)-(12), by solving the characteristic vector of this matrix, obtaining the coefficient matrix B=[β of normalization Legnedre polynomial ¹, β ²..., β ^m], can obtain by the PSWF pulse signal of normalization Legendre approximation by polynomi-als thus;

To obtained PSWF pulse signal, calculate the cross-correlation matrix C of this pulse:

And diagonalization conversion is carried out to this cross-correlation matrix, i.e. X ^tcX=Λ, obtains the transformation matrix X of cross-correlation matrix C diagonalization side.

According to formula D=X ^tb, digital simulation matrix D.

3. the system of polynomials number vector d of orthogonal PSWF pulse is obtained by each row vector of fit metric D _i;

4. according to the normalization Legendre fitting of a polynomial form of orthogonal PSWF pulse: ψ '=DP, wherein P is normalization Legendre multinomial, this completes orthogonal PSWF Pulse Design, as shown in Figure 3.

(2) the logical PSWF orthogonalization design of band

For the logical PSWF orthogonalization design of band, orthogonal PSWF pulses generation flow process disclosed in this invention as shown in Figure 4, can carry out pulses generation as follows.

1. radio frequency channel divides: be that spectrum width channel of communication and orthogonal pulses taken is divided into the identical and mutually overlapping wavelet road of k bandwidth with even overlap mode, k be greater than 0 positive integer, frequency spectrum overlapping size ρ in adjacent wavelet road can account for sub-wave channel bandwidth percentage by the spectral bandwidth that two adjacent wavelet roads are overlapping represents, its span be greater than 0 and be less than 100% percentage;

2. optimum configurations: be the parameter that orthogonal pulses is set, the spectrum width B taken by orthogonal pulses, frequency spectrum lower frequency limit f _lwith frequency spectrum upper limiting frequency f _h, to divide number k, the time-bandwidth product factor c of elliptically spherical function, the overlapping degree of frequency spectrum be that ρ determines in wavelet road, orthogonal pulses spectrum width B and frequency spectrum lower frequency limit f _lwith frequency spectrum upper limiting frequency f _hthree meets relational expression: B=f _h-f _l, orthogonal pulses spectrum width B, sub-band bandwidth B ₀spend ρ three meet relational expression with frequency spectrum is overlapping: B=[(1-ρ) k+ ρ] B ₀, group radio frequency channel mutually overlapping 50% time, each sub-wave channel bandwidth in wavelet road is B ₀=2B/ (k+1); Time-bandwidth product factor c and pulse duration T, each wavelet road bandwidth B ₀three meets relational expression: c=π B ₀t, the elliptically spherical function pulse number in each wavelet road is

3. characterisitic function and integral equation is built: be build characterisitic function and integral equation for each wavelet road, constructed characterisitic function is h _k(t)=2f _k,Hsinc (2f _k,Ht)-2f _k,Lsinc (2f _k,Lt), wherein, f _k,L, f _k,Hbe respectively lower-frequency limit and the upper limit of a kth sub-radio frequency channel, constructed integral equation is wherein, λ _kcharacterisitic function h when the numerical algorithm decomposed for feature based value solves integral equation _kthe characteristic value of (t) constructed matrix, ψ _k(c, t) is λ _kcorresponding characteristic function, the t time variable that to be interval be [-T/2, T/2];

4. solving equation: be the integral equation that the Parr numerical algorithm decomposed by feature based value solves constructed by each wavelet road respectively, and by getting the characteristic function corresponding to a front m eigenvalue of maximum, obtain the elliptically spherical function pulse in each wavelet road;

5. based on the diagonalizable orthogonalization of cross-correlation matrix: referring to the cross-correlation matrix by calculating each wavelet road PSWF pulse participating in orthogonalization design, this cross-correlation matrix being carried out diagonalization conversion, thus achieves the orthogonalization of PSWF pulse.

PSWF ψ is led to the band participating in orthogonalization design and calculates its cross-correlation matrix C:

Diagonalization conversion is carried out to this cross-correlation matrix, i.e. X ^tcX=Λ, obtains the transformation matrix X of cross-correlation matrix C diagonalization side.The transposed form of transformation matrix X is carried out matrix multiple with pulse ψ and obtains orthogonal PSWF pulse ψ ', i.e. ψ '=X ^tψ.Thus achieve the orthogonal design of the logical PSWF of band.

Claims (3)

1. based on the diagonalizable orthogonal PSWF pulse design method of cross-correlation matrix, be a kind of orthogonal PSWF pulse design method, by calculating the cross-correlation matrix C of the PSWF pulse ψ participating in orthogonalization design, this cross-correlation matrix wherein c _i,jbe the cross-correlation function of i-th pulse and a jth pulse, namely this cross-correlation matrix is carried out diagonalization conversion, i.e. X ^tcX=Λ, obtains orthogonal matrix X, and wherein Λ is diagonal matrix, then through the transposed matrix X of orthogonal matrix X ^tmatrix multiple is carried out: X with PSWF pulse ψ ^tψ, thus the orthogonalization realizing PSWF pulse ψ, wherein, PSWF pulse ψ both can be base band PSWF pulse, also can be the logical PSWF of band.

2. according to claim 1 based on the diagonalizable orthogonal PSWF pulse design method of cross-correlation matrix, described base band PSWF pulsed orthogonalization design is: the numerical solution algorithm based on normalization Legendre approximation by polynomi-als obtains the Legendre multinomial weighting coefficient matrix B of PSWF pulse, calculate the cross-correlation matrix C of the PSWF pulse ψ participating in orthogonalization design, this cross-correlation matrix is carried out diagonalization conversion: X ^tcX=Λ, obtains orthogonal matrix X, and wherein Λ is diagonal matrix, then by the transposed matrix X of orthogonal matrix X ^tmatrix multiple is carried out: X with the Legendre multinomial weighting coefficient matrix B of PSWF pulse ^tb, obtains the weighted sum matrix D=X of PSWF pulse ^tb, thus the normalization Legendre fitting of a polynomial form obtaining orthogonal PSWF pulse: ψ '=DP, namely achieve the orthogonalization design of PSWF pulse ψ, wherein P is normalization Legendre multinomial.

3. according to claim 1 based on the diagonalizable orthogonal PSWF pulse design method of cross-correlation matrix, described band leads to the design of PSWF pulsed orthogonalization, and its design procedure comprises: radio frequency channel division, optimum configurations, structure characterisitic function and integral equation, solving equation, based on the diagonalizable orthogonalization of cross-correlation matrix;

Wherein, radio frequency channel divides is that spectrum width channel of communication and orthogonal pulses taken is divided into the identical and mutually overlapping wavelet road of multiple bandwidth with even overlap mode;

Optimum configurations is the parameter arranging orthogonal pulses, the spectrum width B taken by orthogonal pulses, frequency spectrum lower frequency limit f _lwith frequency spectrum upper limiting frequency f _h, to divide number k, the time-bandwidth product factor c of elliptically spherical function, the overlapping degree of frequency spectrum be that ρ determines in wavelet road;

Structure characterisitic function and integral equation are the integral equation definitions building the logical PSWF of band for each wavelet road;

Solving equation refers to that the band adopting Parr numerical algorithm to solve each wavelet road respectively leads to the numerical approximation solution of PSWF;

Refer to based on the diagonalizable orthogonalization of cross-correlation matrix: by calculating the cross-correlation matrix C of each wavelet road PSWF pulse ψ participating in orthogonalization design, this cross-correlation matrix is carried out diagonalization conversion, i.e. X ^tcX=Λ, obtains orthogonal matrix X, and wherein Λ is diagonal matrix, then through the transposed matrix X of orthogonal matrix X ^tmatrix multiple computing is carried out: X with PSWF pulse ψ ^tψ, thus the orthogonalization design realizing PSWF pulse ψ.